This project is designed to use mathematical and computational methods to develop models of individual dendrites, individual neurons, and neuron networks that are based on experimental data. Three projects have been studied during FY1999. Two projects used models systems developed in Mathematica. The first continues a theoretical analysis of why neuronal dendrites branch, based on morphological data obtained earlier in this laboratory. The results support the hypothesis that dendrites branch extensively because they must invade source regions for innervation in order to maximize the amount of synaptic current that can be delivered to the soma. The volume elements within 150 microns of the branches of simulated dendrites are each weighted by the calculated electrotonic connectivity to the soma of the nearest dendrite segment. This coupling-weighted external volume, when summed and divided by the internal dendrite volume, gives a coupled volume ratio (CVR) as the figure of merit to be optimized. Several morphological parameters are adjusted in a search algorithm to generate spatially idealized models whose CVRs are compared to actual motoneurons. Optimal values of the four parameters produced dendrites that resembled those in the target set of 60 dendrites from reconstructed cat motoneurons. We conclude that the ratio of external coupled volume to internal dendrite volume is a realistic figure of merit for neurons, and that motoneuron dendrites come reasonably close to optimal shapes for this figure of merit. The second project continues work on the neuronal interactions in a model of the basic circuit that produces rhythmic respiration that is based on experiments of Dr. Jeffrey Smith. The dynamic behavior of up to 20 interconnected neurons with data-based properties and connections were visualized into computed videos, with excitatory and inhibitory action in the cells modeled as different colored deformations in a flexible sheet and synaptic traffic seen as colored areas flowing from cell to cell. These movies permit visualization of the synchronization of independent oscillators as their interconnection strength increases, as well as the sequence of synaptic flows and membrane dynamics among the five cell types that shape the time course of respiration. A third project was completed in FY1999. We developed two alternative approaches to the construction of equivalent cable models for neuronal dendrites which differ from the conventional cable formulation based on electrotonic distance from the soma. The first alternative uses the outward attenuation of membrane voltage when steady-state currents are delivered to the soma and the other uses the delay of transient somatic voltage perturbations as bases for equivalent cables. Both of these formulations take account of the irregular electrotonic structure of natural dendrites which is lacking in the standard cable model. Both of the new cables provide more accurate simulation of voltage transient behavior of the fully branched model neurons than is given by electrotonic cable models. Computer simulation software was developed in house and written in PASCAL. A full paper is in press. - neurons. morphology, networks, simulations, mathematical models